We define twisted Hochschild homology for Green functors. This construction is the algebraic analogue of the relative topological Hochschild homology THH Cn (−), and under flatness conditions it describes the E 2 term of the Künneth spectral sequence for relative THH. Applied to ordinary rings, we obtain new algebraic invariants. Extending Hesselholt's construction of the Witt vectors of noncommutative rings, we interpret our construction as providing Witt vectors for Green functors.

Abstract. In recent work, Hess and Shipley [18] defined a theory of topological coHochschild homology (coTHH) for coalgebras. In this paper we develop computational tools to study this new theory. In particular, we prove a Hochschild-Kostant-Rosenberg type theorem in the cofree case for differential graded coalgebras. We also develop a coBökstedt spectral sequence to compute the homology of coTHH for coalgebra spectra. We use a coalgebra structure on this spectral sequence to produce several computations.

We show that K2i) is free abelian of rank m − 1. This is accomplished by showing that the equivariant homotopy groups TR n q−λ (Z; p) of the topological Hochschild T-spectrum T (Z) are free abelian for q even, and finite for q odd, and by determining their ranks and orders, respectively.

We describe a construction of the cyclotomic structure on topological Hochschild homology (T HH) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes place entirely in the category of equivariant orthogonal spectra, avoiding use of the Bökstedt coherence machinery. We are also able to define two relative versions of topological cyclic homology (T C) and T R-theory: one starting with a ring Cn-spectrum and one starting with an algebra over a cyclotomic commutative ring spectrum A. We describe spectral sequences computing the relative theory A T R in terms of T R over the sphere spectrum and vice versa. Furthermore, our construction permits a straightforward definition of the Adams operations on T R and T C. ContentsProposition 2.2 ([29, II.4.3]). The category S G of orthogonal G-spectra is equivalent to the category of J G -spaces, i.e., the continuous equivariant functors from J G to T G . The symmetric monoidal structure is given by the Day convolution.This description provides simple formulas for suspension spectra and desuspension spectra in orthogonal G-spectra.Definition 2.3 ([29, II.4.6]). For any finite-dimensional G-inner product space V we have the shift desuspension spectrum functorThis is the left adjoint to the evaluation functor which evaluates an orthogonal G-spectrum at V . Remark 2.4. In [20], the desuspension spectrum F V S 0 is denoted as S −V and F 0 A is denoted as Σ ∞ A in a nod to the classical notation. (They write S −V ∧ A for F V A ∼ = F V S 0 ∧ A.

We study the homology of free loop spaces via techniques arising from the theory of topological coHochschild homology (coTHH). Topological coHochschild homology is a topological analogue of the classical theory of coHochschild homology for coalgebras. We produce new spectrum-level structure on coTHH of suspension spectra as well as new algebraic structure in the coBökstedt spectral sequence for computing coTHH. These new techniques allow us to compute the homology of free loop spaces in several new cases, extending known calculations.

We consider the algebraic K-theory of a truncated polynomial algebra in several commuting variables, K.kOEx 1 ;:::;x n =.x a 1 1 ;:::;x a n n //. This naturally leads to a new generalization of the big Witt vectors. If k is a perfect field of positive characteristic we describe the K-theory computation in terms of a cube of these Witt vectors on N n . If the characteristic of k does not divide any of the a i we compute the K-groups explicitly. We also compute the K-groups modulo torsion for k D Z. To understand this K-theory spectrum we use the cyclotomic trace map to topological cyclic homology, and write TC.kOEx 1 ;:::;x n =.x a 1 1 ;:::;x a n n // as the iterated homotopy cofiber of an n-cube of spectra, each of which is easier to understand.

We describe a construction of the cyclotomic structure on topological Hochschild homology (T HH) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes place entirely in the category of equivariant orthogonal spectra, avoiding use of the Bökstedt coherence machinery. We are also able to define two relative versions of topological cyclic homology (T C) and T R-theory: one starting with a ring C n -spectrum and one starting with an algebra over a cyclotomic commutative ring spectrum A. We describe spectral sequences computing the relative theory over A in terms of T R over the sphere spectrum and vice versa. Furthermore, our construction permits a straightforward definition of the Adams operations on T R and T C.

We show that the relative algebraic K-theory groupis free abelian of rank 1 and that2 . We also find the group structure of K 2i+1 (Z[x, y]/(xy), (x, y)) in low degrees.